On Exponential Stability of Wonham Filter

In [1], Atar and Zeitouni established an exponential asymptotic (in t → ∞) rate of convergence for the distance between differently initialized filter outputs when a signal is Markov process valued in a finite state space for both the discrete and continuous time cases (see related topic in [2]). In this note, we discuss only continuous time case1 and deal with exponential stability of the Wonham filter. Following [1], we assume that a signal X = (Xt)t≥0 is homogeneous Markov process, valued in a finite alphabet S = {a1, . . . , ad}, with transition intensities matrix Λ (with entries λij) and initial distribution vector ν. An observation Y = (Yt)t≥0 is defined as